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Lattice QCD: current status and future
perspectives
C. Rebbi
Physics Department
and
Center for Computational Science
Boston University
Cornell University, February 27, 2008
A brief overview of lattice QCD
QCD: quark fields ψc,s,f (x) and gluon fields Aµ,i(x)
Aµ(x) =∑
i Aµ,iλi, ψ(x)eıgR y
x Aµ(x) dxµψ(y)
Wilson 1974: discretize the theory on a space-time Euclidean lattice
and use as fundamental gauge variables the finite group elements
Uµ(x) = eıgR x+µa
x Aµ(x) dxµwhere a is the lattice spacing.
Quantum mechanical expectation values: 〈O〉 = Z−1×∫ ∏
x,µ dUµ(x)∏
x(dψ(x) dψ(x))O(ψ, ψ, U)e−Sg(U)−ψD(U)ψ
Sg(U) discretizes∫
d4x 14FµνF µν
ψD(U)ψ discretizes∫
d4x ψ(x)[∑
µ γµ(∂µ + ıgAµ) + m]ψ(x)
Integrate over ψ, ψ explicitly: 〈O〉 = Z−1×∫ ∏
x,µ dUµ(x)〈O〉Ue−Sg(U)Det[D(U)]
Approximate the integral over U by stochastic simulation.
From 〈O〉 derive experimentally measurable quantities, e.g.
consider
O = ψ(0)Γψ(0)∑
#x ψ(%x, t)Γψ(%x, t)
ψψΓψ ψΓψ ψψΓ Γψ
〈O〉U 〈O〉
〈O〉 ≡ G(t) =∑
n |〈0|ψΓψ|n, %p = 0〉|2e−mnt
Example with Γ = γ5 → pseudoscalar spectrum:
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60 70
G(t/a
)
t/a
Pseudoscalar, am1=0.0800, am2=0.1000
100 confs.average
(from a work with R. Babich, F. Berruto, N. Garron, Ch. Hoelbling,
J. Howard, L. Lellouch, N. Shoresh)
Formidable calculations: a lattice of size 503 × 100 gives origin to a
quark field with 503 × 100× 12 = 150M degrees of freedom and
to a discretized Dirac operator given by a 150M× 150M (sparse)
matrix.
DetD(U) cannot be stochastically sampled (not directly), rather
reformulate:∫ ∏
dU〈O〉UE−Sg(U)Det[D(U)] =∫ ∏dU
∏dπU
∏dφe−SG(U)−π2
U−φ∗D(U)−1φ
the whole set U,πU , φ is then sampled by a stochastic simulation
which uses a Hamiltonian evolution of U,πU through group space
as an intermediate step (hybrid Monte Carlo calculation); this
requires multiple solutions of D(U)χ = φ.
Additional complication: a naive discretization of the Dirac equation∑
µ γµ(∂µ + ıgAµ) →∑
µUµ(x)ψ(x+µa)−U†
µ(x−µa)ψ(x−µa)2a
introduces 15 extra fermion modes.
Solve by: - removing the extra modes with counterterms (Wilson
fermions, breaks chiral invariance)
- splitting the fermion Dirac components through the lattice
(staggered fermions, reduces the number of fermionic modes from
16 to 4, keeps one non-singlet U(1) chiral symmetry)
- introducing a fifth dimension or equivalently trading the sparse
Wilson Dirac matrix DW (U) for a much more challenging operator
DO(U) = I + DW (U)[DW (U)†DW (U)]−1/2 (domain wall or
overlap fermions, no extra modes and chiral symmetry is preserved
at the cost of much more demanding calculations.)
Forefront calculations in the U.S.
http://www.usqcd.org/
With an effort that started in 1999 the U.S. lattice community, led by
Bob Sugar and the USQCD Executive Committee (R. Brower,
M. Creutz, N. Christ, P. Mackenzie, J. Negele, C. R., D. Richards,
S. Sharpe), has organized itself in a national USQCD collaboration.
With DOE SciDAC support the collaboration has produced a suite of
software tools which help writing code for efficient lattice QCD
simulations, with DOE HEP and NP support it has deployed
dedicated computers for LQCD calculations, and it has been able to
secure large allocations of supercomputer time at the DOE facilities
managed by DOE ASCR.
The dedicated resources are allocated by the collaboration’s
Scientific Program Committee and are used for large scale projects,
which generate sets of gauge field configurations available to the
whole collaboration, as well as for medium and small scale projects.
Examples of available or planned configurations by the MILC
collaboration (with staggered fermions):
a(fm) ml/ms size L(fm)
0.09 0.10 403 × 96 3.6
0.09 0.05 563 × 96 5.0
0.06 0.05 843 × 144 5.0
0.06 1/27 1003 × 144 6.0
0.045 0.05 1123 × 192 5.0
0.045 1/27 1243 × 192 5.6
In 2007 the USQCD Executive Committee, with major contributions
also from T. Applequist, C. Bernard, S. Catterall, C. DeTar,
G. Fleming, J. Hetrick, F. Karsch, R. Mawhinney, C. Morningstar,
R. Narayanan, H. Neuberger, K. Orginos, M. Savage, M. Schmaltz,
M. Unsal and P. Vranas, has produced four White papers illustrating
near term goals for lattice QCD calculations (see
http://www.usqcd.org/collaboration.html#2007Whitepapers).
The following areas have been identified:
- Fundamental constants and hadronic matrix elements for
electroweak interactions.
- QCD thermodynamics.
- Hadron spectrum, structure and interactions.
- Strong dynamics for physics beyond the standard model.
Illustrative results (data from the Fermilab Lattice, HPQCD and
MILC collaborations, compilation due to P. Lepage):
0.9 1 1.1
LQCD/Exp’t (nf = 0)LQCD/Exp’t (nf = 0)
0.9 1 1.1
LQCD/Exp’t (nf = 3)LQCD/Exp’t (nf = 3)
Υ(1P − 1S)Υ(3S − 1S)Υ(2P − 1S)Υ(1D − 1S)ψ(1P − 1S)
Mψ −Mηc
MD∗s−MDs
2MBs −MΥ
2MDs −Mηc
3MΞ −MN
MΩ
fK
fπ
α(5)
MS(MZ)
Determinations of αs (αL obtained by the HPQCD collaboration):
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
q2/mDs*2
0
0.5
1
1.5
2
2.5
f +(q2 )
0
experiment [Belle, hep-ex/0510003]lattice QCD [Fermilab/MILC, hep-ph/0408306]
D → Klν
Form factor for semileptonic D decay, from the Fermilab/MILC
collaboration.
0
2
4
6
8
10
100 200 300 400 500 600 700
0.4 0.6 0.8 1.0 1.2 1.4 1.6
T [MeV]
Tr0 (ε-3p)/T4
p4: Nτ=46
asqtad: Nτ=46
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
p/ε
T/T0
S/Nq=1015
100p4fat3, Nq=0asqtad, Nq=0
Results for the QCD equation of state, from the MILC collaboration
and Ejiri, Karsch, Laermann, and Schmidt.
Strange quark contribution to the nucleon form factors
Research in progress, in collaboration with R. Babich, R. Brower,
M. Clark, G. Fleming, and J. Osborn
sΓs
NN
Partial results based on 386 gauge field configurations, with two
flavors of dynamical quarks (Wilson discretization), on anisotropic
243 × 64 lattices (as = 0.108(7)fm, at = 0.036(2)fm),
generated by the LHPC collaboration.
We must calculate 〈N |ψsΓψs|N〉 =
limt→∞
P!x,!y eı(!p·!x+!q·!y)〈ψψψ(#x,t) ψsΓψs(#y,0) ψψψ(#0,−t)〉
P!x〈ψψψ(#x,t) ψψψ(#0,−t)〉 − 〈ψsΓψs〉
For each gauge field configuration we calculate separately the
propagators Pcs,c′s′(U ; x, y) which solve
[D(U)P (U)]cs(x) = δ(x, y)δc,c′δs,s′
for the light and strange quarks. We combine the light quark
propagators into nucleon propagators (with implicit sums over color
and spin indices)∑
#x eı#p·#xεc1,c2,c3εc′1,c′2,c′3Ψ∗
s1,s2,s3Ψs′1,s′2,s′3
Pc1s1,c′1s′1(U ; %x, t,%0,−t)
Pc2s2,c′2s′2(U ; %x, t,%0,−t)Pc3s3,c′3s′3
(U ; %x, t,%0,−t)
and the strange quark propagators into∑#y eı#q·#yTr[P (U ; %y, 0, %y, 0)Γ]
and average the product of the two over the gauge field
configurations.
Note that this would require the calculation of 243 = 13, 824
strange quark propagators per configuration (prohibitive). Various
techniques can be used to reduce the computational cost. We
calculate the strange quark propagators with 64 sources staggered
through the lattice, separated by 16at in time and 6√
3as in space.
We have verified that the non-diagonal contributions largely cancel
because of the fall off of the propagators and gauge averaging. The
number of propagators which we calculate per configuration is 864.
Other computational improvements are under study.
(Different computational approaches are followed by other groups,
cfr. S.J. Dong, K.-F. Liu, and A. G. Williams, 1998, R. Lewis,
W. Wilcox and R. M. Woloshyn, 2003.)
Current results for the zero-momentum scalar density∑#x ψsψs(%x, 0), showing also the effect of decorrelating scalar
density and nucleon propagators.
-0.5
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12 14 16
R S
(t - t’)/a = (t’- t0)/a
correctshifted
The contributions to ψsψs from spatial points far away from the
nucleon source add noise but very little signal. The error can be
reduced by windowing (although this introduces some momentum
contamination).
-0.5
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12 14 16
R S
(t - t’)/a = (t’- t0)/a
all (243)133
Current results for the zero-momentum axial-vector current∑#x ψsγ5γiψs(%x, 0), showing also the effect of windowing.
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0 2 4 6 8 10 12 14 16
R A
(t - t’)/a = (t’- t0)/a
all (243)133
Conclusions
- Lattice calculations provide very accurate results for a variety of
QCD observables which cannot be otherwise evaluated.
- The calculation of many observables presents formidable
computational challenges.
- With increasing computational power and progress in algorithms
the accuracy and scope of lattice calculations continue to increase.